• Skip to primary navigation
  • Skip to main content
  • Skip to primary sidebar
The Analysis Factor

The Analysis Factor

Statistical Consulting, Resources, and Statistics Workshops for Researchers

  • Home
  • About
    • Our Programs
    • Our Team
    • Our Core Values
    • Our Privacy Policy
    • Employment
    • Guest Instructors
  • Membership
    • Statistically Speaking Membership Program
    • Login
  • Workshops
    • Online Workshops
    • Login
  • Consulting
    • Statistical Consulting Services
    • Login
  • Free Webinars
  • Contact
  • Login

When to Use Logistic Regression for Percentages and Counts

by Karen Grace-Martin 1 Comment

One important yet difficult skill in statistics is choosing a type model for different data situations. One key consideration is the dependent variable.

For linear models, the dependent variable doesn’t have to be normally distributed, but it does have to be continuous, unbounded, and measured on an interval or ratio scale.

Percentages don’t fit these criteria. Yes, they’re continuous and ratio scale. The issue is the boundaries at 0 and 100.

Likewise, counts have a boundary at 0 and are discrete, not continuous. The general advice is to analyze these with some variety of a Poisson model.

Yet there is a very specific type of variable that can be considered either a count or a percentage, but has its own specific distribution.

This is a variable that indicates the number of successes out of N trials.

Here are a few examples that I’ve seen recently in consulting:

  • The percentage of buds on a tree that opened.
  • The number of errors a bilingual speaker made on a grammar test.
  • The percentage of employees a manager would recommended for a promotion under different conditions.

If you think about it, you can consider any of these to be either a percentage or a count.

If a tree has 820 buds and 453 open, we could either consider that a count of 453 or a percentage of 55.2%. But they’re both measuring this same idea of 453 out of 820 possible openings.

There isn’t a name for it, but I think of this type of percentage as a discrete percentage.

It’s a little bit different than a percentage of a mass quantity, like the percentage of the area of a Petri dish that is covered with mold. In the Petri dish example, there aren’t discrete trials, each of which could be a success or a failure. It’s just one mass and some measurable percentage of it is covered with mold.

A variable that measures the number of success out of N trials follows a binomial distribution, which is simply the sum of a set of success/failure trials.

The binary outcome variable that we generally use for logistic regression is one of these trials. It follows a Bernoulli distribution. The variable has one trial with two possible outcomes for each individual: success or failure.

(Yes, this is very confusing. Many books, software, and statisticians describe the one-trial binary situation as a binomial distribution. I’ve surely done it myself. They’re related but not exactly the same).

As it turns out, logistic regression can handle either a Bernoulli variable with one trial per subject or a Binomial variable with N trials per subject.

The key parameter in both distributions is p, the probability of success on each trial.

That’s what we’re trying to predict in a logistic regression with our predictor variables. For example, does the tree’s altitude predict the probability that any given bud will open?

Pretty much every stat software has both options as dependent variables for a logistic regression, but it’s not always easy to find. For example, in SAS, it’s quite easy. The MODEL statement in PROC LOGISTIC allows either. It calls them the single-trial syntax or the events/trials syntax.

But in SPSS, the Logistic Regression procedure can only run the single-trial Bernoulli form. To run the events-and-trials binomial form, you need to use the Generalized Linear Models procedure. There you can specify a logistic link and a binomial distribution.

Now, why not treat it as a count variable and run a Poisson or negative binomial model?

Well….there is a relationship between the binomial and the Poisson distributions: as the number of trials gets big and the probability of success gets small, the binomial distribution approximates a Poisson.

So think about how many trials you actually have and the overall proportion of successes to decide which approach might fit better.

Binary, Ordinal, and Multinomial Logistic Regression for Categorical Outcomes
Get beyond the frustration of learning odds ratios, logit link functions, and proportional odds assumptions on your own. See the incredible usefulness of logistic regression and categorical data analysis in this one-hour training.

Tagged With: binomial, Count data, count model, dependent variable, events, logistic regression, Negative Binomial Regression, percentage data, Poisson Regression, trials

Related Posts

  • Member Training: Count Models
  • When Dependent Variables Are Not Fit for Linear Models, Now What?
  • Proportions as Dependent Variable in Regression–Which Type of Model?
  • Poisson Regression Analysis for Count Data

Reader Interactions

Comments

  1. Meghan Caulfield says

    February 3, 2020 at 10:10 am

    If you interpret the odds ratio from a logistic regression of one trial as the relative odds between two groups, how do you interpret the odds ratio from a logistic regression of N trials per subject? Thanks.

    Reply

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Please note that, due to the large number of comments submitted, any questions on problems related to a personal study/project will not be answered. We suggest joining Statistically Speaking, where you have access to a private forum and more resources 24/7.

Primary Sidebar

Free Webinars

Effect Size Statistics on Tuesday, Feb 2nd

This Month’s Statistically Speaking Live Training

  • January Member Training: A Gentle Introduction To Random Slopes In Multilevel Models

Upcoming Workshops

  • Logistic Regression for Binary, Ordinal, and Multinomial Outcomes (May 2021)
  • Introduction to Generalized Linear Mixed Models (May 2021)

Read Our Book



Data Analysis with SPSS
(4th Edition)

by Stephen Sweet and
Karen Grace-Martin

Statistical Resources by Topic

  • Fundamental Statistics
  • Effect Size Statistics, Power, and Sample Size Calculations
  • Analysis of Variance and Covariance
  • Linear Regression
  • Complex Surveys & Sampling
  • Count Regression Models
  • Logistic Regression
  • Missing Data
  • Mixed and Multilevel Models
  • Principal Component Analysis and Factor Analysis
  • Structural Equation Modeling
  • Survival Analysis and Event History Analysis
  • Data Analysis Practice and Skills
  • R
  • SPSS
  • Stata

Copyright © 2008–2021 The Analysis Factor, LLC. All rights reserved.
877-272-8096   Contact Us

The Analysis Factor uses cookies to ensure that we give you the best experience of our website. If you continue we assume that you consent to receive cookies on all websites from The Analysis Factor.
Continue Privacy Policy
Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Necessary
Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.